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Mathematical Charts and Tables

Charts and Tables for Instruction

Charts and Tables for Instruction

Ancient Babylonian scribes learned to record numerical tables on clay tablets. If a schoolroom or library burned, the tablets baked hard, and might survive for millennia. Several such tablets are preserved in libraries today, and the replica of one of these at the Smithsonian reminds us of the long history of these objects.

In the 19th century, new European ideas about teaching arithmetic to very young children reached the United States. In 1831, the Boston firm of Munroe & Francis published a series of some fifty “infant school cards,” designed to teach subjects ranging from arithmetic to reading to natural history. Teachers were to use the arithmetic cards in conjunction with another piece of apparatus newly introduced in Western Europe and then in the United States, the teaching abacus or numeral frame.

Charts also were used to teach about weights and measures. In the wake of the French Revolution of 1789, the French developed an entirely new system of measuring distance, area, volume, temperature, and even time. By the 1860s, several European countries had adopted a revised version of this metric system, and metric weights and measures were legalized in the United States. An organization known as the American Metric Bureau began to distribute metric demonstration apparatus for the classroom. In the 1890s, the American Metric Bureau began to sell a metric chart for educational use.

Within the decade, the metric system was but one of several topics illustrated in a set of charts copyrighted by R. O. Evans of Chicago. Evans’ set of twenty charts illustrated such wide ranging topics as counting and writing numbers, arithmetic operations, fractions, the area of surfaces and the volumes of solid, business methods, and surveying.

In the first half of the 20th century, machines that could do ordinary arithmetic became common in the store and the office, and inexpensive adders were available for consumers. In the years following World War II, educators placed new emphasis on understanding the principles underlying arithmetic. Charts such as number lines sold for classroom use.

In the 19th century, Americans began to teach young groups of children in classrooms. Some were designed especially for these children, and were called infant schools. To create a vivid impression on young minds, teachers used a numeral frame or abacus in combination with a chart like this one.

This cardboard chart was part of a larger series. It has printing on both sides. It is labeled on one side: ARITHMETIC CARD II. This side shows a group of common objects on the left, and one of these objects on the right. It was designed to teach adding 1 to 6, 7, 8, and 9. Teachers were told to perform the same operation using balls on an abacus. The other side of this chart is entitled: ARITHMETIC CARD VI. It has groups of vertical lines on the left and two slanting lines on the right, and was meant to teach subtraction of 2. It also was to be used with an abacus.

A mark on the chart reads: INFANT SCHOOL CARDS, PUBLISHED BY MUNROE & FRANCIS, BOSTON. For another chart in the series, see CL*389116.04.

Infant schools were popular in Boston around 1830, and the abacus was introduced into the Boston schools at about that time. Munroe & Francis was in business from the last decades of the 1700s until 1860 or so. In October 1831, The New England Magazine announced that Munroe and Francis had just published “Complete Sets of Lessons on Cards for Infant Schools, consisting of 100 Lessons of every variety, on 50 Boards.” It seems likely that these cards were part of that set.

In the wake of the Revolution of 1789, French scientists developed a new system of weights and measures known in English-speaking countries as the metric system. A handful of early 19th-century American mathematics textbooks discussed metric measurements. In the 1860s, metric measures were legalized in the United States, although they were not mandatory. A few advocates of the new system, most notably the distinguished librarian Melville Dewey, joined together to form the American Metrological Society and to advocate the use of metric measures. This chart was prepared by the Society for classroom use.

The tan paper chart shows a meter length divided into decimeters, centimeters, and millimeters. It also shows a liter container and a block 1,000 cubic centimeters in size. It gives the value in United States currency of silver coins weighing from 1 gram to 1000 grams. A mark at the bottom front of the chart reads: Copies of this chart will be mailed on receipt of ten cents in postage stamps. (/) ADDRESS AMERICAN METROLOGICAL SOCIETY, 41 EAST 49TH ST., N.Y. CITY.

Science magazine noted publication of the chart in 1891, which is used as the approximate date of the object. This example was found uncatalogued in the collections of the Smithsonian Institution Libraries.

Around 1900 many American educators advocated the use of objects in teaching mathematics and the sciences. R. O. Evans Company of Chicago published this set of twenty chromolithographed charts. They were designed to apply the object method “to the entire subject of practical arithmetic.” The title chart shows a man in classical garb holding a diagram of the Pythagorean theorem and a pair of dividers, expounding to a child. Other instruments displayed include a pencil, a drawing pen, a magnetic compass, several geometric models, a globe, a telescope, two set squares, an hourglass, and one of Evans’s charts.

Charts include extensive commentary for teachers. There are sheets entitled Counting and Writing Numbers, Reviews and Colors, Addition, Subtraction, Multiplication, and Division,. Other charts discuss Fractions, Weights and Measures, the Metric System, and Mensuration (one chart considers the measurement of flat surfaces, another one 3-dimensional solids). There also are charts on Business Methods (3 charts), Lumber and Timber Measure,Surveying, Percentage, Commercial and Legal Forms, and Book Keeping. A variety of objects are shown.

The paper, cloth-backed charts are held together at the top by a piece of fabric that is tacked to a wooden backing. This backing slides into an oak case decorated with machine-made molding and panels. This particular example of Evans’ Arithmetical Study was used at a school in New Hampshire.

Both written language and written tables originated in the ancient Middle East. Scribes kept lists of numerical data, such as the number of sheep and goats transferred on different days of the month. A few of the clay tablets on which they wrote survive to this day. A tiny number of these tablets have rows and columns arranged in tables.. The rows may give totals of number of various forms of livestock transferred over time, with a column for the animals that were the responsibility of each person charged with such matters. Such documents date from around 2020 BCE.

Those learning and teaching mathematics in ancient Iraq rarely displayed information in tabular form. However, in 1922 the American collector George Plimpton purchased such a tablet. This replica of that unusual object was made in 1957 by L. C. Eichner. Plimpton donated the original object to Columbia University in the 1920s. The original dates from about 1800 BCE, and reportedly was excavated in what is now Iraq at the side of the ancient city of Lasra. The portion of the tablet that survives has four columns of numbers written in the sexagesimal (base 60) system of numbers.

Otto Neugebauer and A. J. Sachs offered a modern mathematical interpretation of the tablet in 1945. They noted that the numbers in the second and third columns of the table might represent the squares of the length of the shortest side and of the hypotenuse of right triangles, and interpreted the table as relating to Pythagorean triples. As the name Pythagorean suggests, such numbers had previously only been associated with later Greek mathematics. Other scholars have suggested that this was a part of a larger table of reciprocal numbers and related geometric figures, compiled by a teacher wishing to have examples of such reciprocals available for use in assignments.

References:

A. Aaboe, Episodes from the Early History of Mathematics, New Haven: Yale University Press, 1964, pp. 30–31.

From the time of Descartes (1596–1650), mathematicians have described positive and negative integers as evenly spaced points on a line, now called the number line, that extends infinitely in both directions. This usage had made it into some school textbooks by the early 20th century. Particularly at the time of the development of the New Math in the 1950s and 1960s, number lines became part of the school classroom. This example of a number line was developed by Loraine McMillan and sold by Houghton Mifflin Company to accompany the 1972 edition of the textbook Modern School Mathematics. McMillan also prepared a leaflet describing how the number line should be used and a that sold separately.

The device consists of eleven cards. Ten of these can be placed end to end to show a number line with the integers from 0 to 100 written in red. The eleventh card is divided into segments but has no numbers marked on it. Each card, unfolded, measures 89 cm. w. x 11 cm. d. The cards were coated with clear plastic so that teachers could mark them with crayons or felt tip markers. The teacher’s guide is printed on blue paper. A mark on it reads: Teacher’s number line; teacher’s guide(/) by (/) Loraine McMillan. Another mark on it reads: houghton (/) mifflin (/) company. A third mark reads: 1972 .

This example appears unused. It was received in 2012, and had been the property of Harvard University mathematician Andrew Gleason.